Novel Method

: Plithogenic decision-making models are evolved integrating the Plithogenic modelling approach with various methods of multi-criteria decision-making (MCDM). The earlier Plithogenic based decision methods are primarily based on the degrees of appurtenance. This paper introduces a novel Plithogenic ranking genre of decision-making paradigm based on degrees of contradiction. The method of Decision Making on Plithogenic Contradictions (DMPC) developed in this research work is indigenous and unique as the modeling procedure doesn’t resemble any of the decision methods. This simple and logical approach proposed in this paper is applied in making optimal decisions on supplier selection. The proposed contradiction based Plithogenic model shall be integrated with other decision methods and this will certainly create a breakthrough in framing contradictions based combined Plithogenic decision-making models.


Introduction
The everlasting conflict of choosing the optimal alternatives satisfying all the criteria to the expected extent is motivating the researchers to develop new methods.This has led to the expansion of the theoretical aspects of decision-making with the development of scientific and algorithmic approaches to decision-making methods.The construction of any decision-making problem comprises certainly an elementary decision-making matrix with values matching the alternatives and criteria.The two prime objectives of the decision methods are to find the criterion weights and ranking of the alternatives.The decision-making methods are classified based on information availability, decision timeline, domain, level, structure, outcome, approach, and process.
The circumstances of making decisions are influenced by several factors affecting the deterministic nature of decision-making.The representations using crisp sets are replaced with the extension of fuzzy sets developed by Zadeh [1] to handle impreciseness and uncertainty.These fuzzy sets are further extended to intuitionistic sets [2] and neutrosophic sets to deal the situations of decision-making with hesitancy and indeterminacy.The decision-making methods developed in crisp sense are discussed by the researchers in the extended version of sets.However, these different representations of set are unified under one roof of Plithogeny by Smarandache [3] in the year 2018.The origin and development of Plithogenic sets has made novel plithogenic decision-making methods to evolve.Smarandache has contributed a lot to the field of Plithogeny, especially to the development of fundamental concepts of the Plithogenic sets [4][5][6].Smarandache has also contributed An International Journal on Informatics, Decision Science, Intelligent Systems Applications

Martin et al., A Novel Method of Decision Making Based on Plithogenic Contradictions
to Plithogenic algebraic structures [7][8].Nivetha and Smarandache have together initialized the conceptualization of Plithogenic based hypergraphs and super hypergraphs [9][10].
A plithogenic set is basically a 5-tuple set that deals with attributes.This set comprises attribute values, degrees of appurtenance, and contradiction.The degrees of appurtenance decide the nature of the Plithogenic sets and it assumes any of the set representations such as crisp, fuzzy, intuitionistic, and neutrosophic.The Plithogenic decision-making methods primarily involve plithogenic operators to obtain a unified decision-making matrix based on the expert's opinion.The literature on Plithogenic based multi criteria decision making (MCDM) methods is limited.Some of the most commonly applied conventional decision-making methods are discussed in Plithogenic environment only with the inclusion of the Plithogenic operators of union and intersection and degrees of appurtenance.This has motivated the authors to develop a new genre of decision-making method based on the degrees of contradiction.The method of making decisions with a contradiction degree is proposed as a method of ranking the alternatives.This method is very simple in its formulation and the logical approach makes the method more rational.
The paper is organized as follows: section 2 sketches out the contributions in the domain of Plithogenic decision-making.Section 3 presents the proposed method of Decision Making on Plithogenic Contradictions.Section 4 applies the proposed method to the supplier selection problem.Section 5 discusses the results under different cases and section 6 concludes the work with future directions.

Literature review
The theory of Plithogeny is applied in MCDM integrating a wide range of different concepts of soft sets, Hypersoft sets, cognitive maps, hypergraphs, and many others.Plithogenic decision-making models are developed based on these concepts to design solutions to real-life problems.Plithogenic based MCDM are either the extensions or the generalizations of the existing mathematical concepts.The Plithogenic logic, probability statistics, and optimization assist in obtaining optimal solutions to decision-making problems.The contributions of researchers towards the formulation of Plithogenic decision-making models are presented in Table 1.In the above mentioned Plithogeny based decision-making methods, the following research gaps are identified.
 The plithogenic operators based on degree of appurtenance are widely applied and only in few instances the contradiction degree is used. The plithogenic oriented decision-making methods lack the use of the aspect of contradiction degree in handling the alternatives and criteria.Hence this research work designs a decision-making method purely based on the contradiction degrees with respect to the dominant attribute value of the alternatives.The novel attributes of this paper are as follows:  A distinctive decision making approach based on contradictions degree. Simple and compatible method of finding the optimal alternatives. Flexible method which accommodates several alternatives and criteria.

Proposed Method of Decision making based on Plithogenic Contradictions
This section consists of the steps involved in the method of Decision Making on Plithogenic Contradictions (DMPC).The elementary steps of this method are similar to the general working principle of an MCDM method.Figure 1 presents the overall framework of the proposed method of DMPC.
The decision making matrix is with m alternatives and n criteria.
Step 2: Finding the Criterion Weights The criterion weights say Wk are determined using any of the methods.Each of the criterion has criterion values say Cki.

Step 3: Construction of contradiction matrix
The dominant criterion value say CkD among the criterion values of each criteria is identified.The contradiction degree among the criterion values is determined.Based on the contradiction degree, the contradiction matrix is constructed with contradiction degrees pertaining to the dominant criterion value with respect to the values assumed by each alternative with respect to the criterion value in the initial matrix.
Step 4: Weighted contradiction matrix The weighted contradiction matrix [WC D ]is obtained by multiplying the criterion weights with the values of contradiction matrix.
Step 5: Finding the score values The score values of each of the alternative with respect to both benefit and cost criteria say BSj and CSh is first calculated.The difference between the values is determined, say BSj -CSh = Df Step 6: Ranking of the alternatives The alternatives are ranked based on the difference values Df.The alternative with maximum difference value is ranked first and so on.

Application of DMPC in supplier selection
In this section, a decision-making problem is solved using the proposed method of DMPC.Let us consider a logistic supplier selection problem with five alternatives and four criteria say C1 -Price, C2 -Time span of delivery, C3 -Flexibility, and C4 -Reliability.
The criteria C1 and C2 are considered to be cost criteria and the criteria C3 and C4 are considered as benefit criteria.
Each criteria presumed to be the attribute possess the attribute values of {L, M, H} i.e. {Low, Moderate, High}.
For the cost criteria, the dominant attribute value is certainly LOW & for the benefit criteria it is HIGH.The initial decision making matrix with linguistic values is presented in Table 2.

Alternatives
The assumed criterion weights and the dominant attribute value with respect to each of criterion are presented as follows in Table 3.
The weighted contradiction matrix is computed using step 4 as follows in Table 5.The differences between the benefit and the cost criteria score values are presented in Table 7.Based on the difference values the alternatives are ranked as follows as in Table 8.
Table 8.Ranking of the alternatives.

Discussion
The above ranking of the alternatives is obtained with assumed criterion weights.The same ranking procedure based on contradictions is repeated with different criterion weights obtained using various methods such as the Analytical Hierarchy Process (AHP), Entropy, and the method of CRITIC (CRiteria Importance through Intercriteria Correlation).Table 9 and Figure 2 represent the rankings of the alternatives using different criterion weights.

Conclusions
This research work proposes a new genre of Plithogenic based decision-making method based on contradictions.The proposed method stands distinct in comparison with other methods as it streamlines a new modality of making optimal decisions.This method will definitely lessen the hurdles in choosing the alternatives based on cost and benefit criteria.The ranking obtained using the Plithogenic method based on contradictions is compared with different criterion weights.This method shall be dealt with extended Plithogenic sets.Also, the method of Plithogenic Cognitive Maps shall be associated with the proposed method as a means of developing several hybrid decisionmaking methods.This method is highly adaptable and flexible in nature and hence it shall be blended with other decision-making models to evolve new hybrid decision-making systems.

Figure 2 .
Figure 2. Graphical representation of diverse ranking of the alternatives.

Table 1 . Contributions of Plithogenic based decision making. Authors & Year Plithogenic Decision Making Method Domain of Application Highlights of the contribution
ranking An International Journal on Informatics, Decision Science, Intelligent Systems Applications Martin et al., A Novel Method of Decision Making Based on Plithogenic Contradictions  Plithogenic aggregation of weights Martin et al. [28] PHS,DM Covid-19  Extended combined plithogenic hypersoft sets An International Journal on Informatics, Decision Science, Intelligent Systems Applications Martin et al., A Novel Method of Decision Making Based on Plithogenic Contradictions An International Journal on Informatics, Decision Science, Intelligent Systems Applications Martin et al., A Novel Method of Decision Making Based on Plithogenic Contradictions An International Journal on Informatics, Decision Science, Intelligent Systems Applications Martin et al., A Novel Method of Decision Making Based on Plithogenic Contradictions

Table 3 .
Decision matrix with criterion description.
The score values of the benefit and cost criteria with respect to each alternative are calculated as presented in

Table 6 .
An International Journal on Informatics, Decision Science, Intelligent Systems Applications et al., A Novel Method of Decision Making Based on Plithogenic Contradictions

Table 6 .
Score values of criteria.

Table 7 .
Difference in score values.

Table 9 .
Ranking of alternatives based on different criterion weights.An International Journal on Informatics, Decision Science, Intelligent Systems Applications et al., A Novel Method of Decision Making Based on Plithogenic Contradictions